Algebra 2 12.1 – 12.4
Sequences & Series
12.1-12.3 Arithmetic and Geometric Sequences
__________________ - an ordered collection of numbers. It’s a function whose domain
is a set of consecutive integers. The values in the range are called the terms of the
sequence. A sequence can either be finite or infinite.
Finite sequence 2, 4, 6, 8
(IT ENDS!)
Infinite sequence 2, 4, 6, 8, …
(DOESN’T END)
Write the 1st 5 terms of an = 2n + 3.
a1 = 2(1) + 3
a2 = 2(2) + 3
a3 =
a4 =
a5 =
If there is a pattern to the sequence, you may be able to write a rule for the nth term.
1 1
1 1
Ex. 1 (  , ,
, ,...)
3 9 27 81
1
=
The next term is _________
Ex. 2
a)
b)
c)
d)
2
3
 1  1  1
 1
(  ,   ,   ,  
 3  3  3
 3
4
So, an = ______________
Choose the general equation. (2, 6, 10, 14, …)
y = 3x - 1
y = 2x + 4
(Hint: Use the table of values on calculator)
y=x+2
y = 4x – 2
Arithmetic sequence – difference between consecutive terms is a constant.
General Rule for an Arithmetic Sequence:
where …
a1 is the __________ term
d is the _____________________
n is the _____________________
an is the value of the _________ term
an  a1  (n  1)d ,
Decide whether each sequence is Arithmetic, or neither. If it is Arithmetic, find
using
an  a1  (n  1)d .
1.
1 1 1 1
, , , . . . A or
2 3 4 5
n
2.
a10
−4, 4, 12, 20, 28 . . . A or n
Geometric sequence – there’s a common ratio between terms.
General Rule for a Geometric Sequence:
an  a1 (r ) n 1
where
a1 is the _________ term of the sequence
r is the ______________________
n is the ______________________
an is the _________ term of the sequence
Ex.
1, 2, 4, 8, 16, 32, …
Find
a8 using an  a1 (r ) n 1 .
What is the ratio (r) ? _______
Decide whether each sequence is Arithmetic, Geometric, or neither. If it is Arithmetic or
Geometric, find
a7 by using either an  a1  (n  1)d
3.
1, 2, 6, 24, 120, … A
5.
2, 5, 10, 17, 26, . . .
A
G
G
n
n
or
an  a1 (r ) n 1 .
4.
81, 27, 9, 3, 1, … A
G
n
6.
1,
3
5
, 2,
, 3, . . . A
2
2
G
n
When the terms of a sequence are added, the result is a _________________. A
series can either be finite or infinite.
Finite series 2 + 4 + 6 + 8
Series – we can use

Infinite series 2 + 4 + 6 + 8 + …
which is called ________________ or ______________.
5
 3i
i1
Trick: y= (the expression) Go to the table. Grab the values between 1 and 5. Add
them together.
General Form for the Sum of the First n Terms in an Arithmetic Series:
 a  an 
S n  n 1
,
 2 
where…
a1 is the ________ term
an is the ________term
n is the ___________ of terms
28
Ex. Find the sum of the arithmetic series  (2  4i ) .
i 1
A
B
C
D
14
420
870
1568
General Form for the Sum of the First n Terms of a Geometric Series:
1 r n 
 , r  1
S n  a1 
 1 r 
where
a1 is the _______ term
r is the ______________
n is the __________ of terms
Ex . Consider the following geometric series. Find the sum of the first ten terms.
a) What is the sum of the first ten terms of 4 + 2
b) What is the value of the series below?
k 1
6
+ 1 + ½ + …?
3
4 

k 1  2 
A
B
C
D
7. Find
15
 (3  2i) .
A
or G ?
i 1
16
8. Find
 4(3)i 1 .
i 1
A
or G ?
275
8
729
16
665
8
2059
16
 1 rn 
 a1  an 
sn  n 

 or sn  a1 
 2 
 1 r 
 1 rn 
 a1  an 
sn  n 

 or sn  a1 
 2 
 1 r 
Choose the best answer.
9.
What is the third term of the sequence defined by an = 5n + 7? ________
10.
What is the next term in the sequence? 0, 2, 6, 12, 20 . . . ? ________
11.
Write out the series
3
 (3i  4) ?
i1
__________________________________
12.4 Notes - Infinite Geometric Series
Infinite Geometric Series: the sum of an infinite series that has a _________
An infinite geometric series may have a ___________, depending on the value of the common ratio.
The Sum of an Infinite Geometric Series: S 
a1
,
1 r
where
a1 is the _________ term of the series
r is the ____________________ and ____________
 If |r| > 1, the series _______________________________
Ex 2
Does the infinite geometric series have a sum?

7
3 

n 1  2 
a)
a1  ______
n 1

b)
r  _______
a)_________________
1
 4 

6
n 1
a1  ______

n 1
c)
10(1)
n 1
n 1
r  _______
b)________________
Find the sum of the infinite geometric series if it has one.
4 4
b) 12  4    ...
a)  2(0.1) i 1
3 9
i 1
a1  ______
r  _______
c)_________________
Ex 3

a1  ______
r  _______
a)_________________
a1  ______
r  _______
b)________________
3 9 27
c) 1     ...
2 4 8
a1  ______
A
B
C
D
r  _______
-2
-½
2
The series has no sum.
12.1-12.3 Sequences and Series Homework Day 1
4.
5.
6.
7.
Is the sequence Arithmetic, Geometric or neither?
If it is A or G then find
8. -2, 0, 2, 4, …
10. 40, 10,
a7 by using either an  a1  (n  1)d
A
5 5
, ,… A
2 8
an  a1 (r ) n 1 .
n
9. 3, 6, 12, 24, …
A
G n
G n
11. 4, 7, 12, 19, …
A
G n
G
a1  ____________
a1  ____________
a  ____________
 a  an  1
S n  n 1
an  ____________
 an  ____________
2


an  ____________
12)
or
13)
14)
12.4 Sequences and Series HW Day 2
For #’s 1 – 3, find the sum of the infinite geometric series, if it exists.
1.

11

 
i 1 3  3 

i 1
2.
  3
i 1
i 1
3.
3.
4. Tell whether the sequence -7, -1, 5, 11, 17, 23, … is arithmetic.
7
5. Find the sum of the arithmetic series
 (4  i) .
i 3
6. Tell whether the sequence
2
, 4, 24, 144, 864, … is geometric.
3
Algebra 2
Sequences & Series Review
Ch 12.1-12.4
TAKE OUT ALL YOUR NOTES FROM CHAPTER 12
Write next the formula what it is used to find.
a n  a1  n  1d
an  a1 r 
 a  an 
S n  n 1

 2 
1 r n
S n  a1 
 1 r
n 1



S
a1
1 r
Ques. 1 – 2. Is each sequence arithmetic, geometric or neither?
1. 108, 54, 27, 13.5…
2. 19, 13, 7, 1, -5,…
Ques. 3 – 4. Find the common difference or the common ratio.
3. -1, 4, -16, 64, …
4. 24, 29, 34, 39 …
Question 5. Find a 20 .
5. 5, 13, 21, 29, …
(Hint: Is it arithmetic or geometric?)
Formula_________________
Question 6. Find a10 .
6. 3, 12, 48, 192, …
(Hint: Is it arithmetic or geometric?)
Formula_________________
Questions 7 – 8. Evaluate each sum.
25
7.
10
 2i  1
8.
i 1
 32
i 1
i 1
Question 9. Find the sum of the first ten terms.
9. 11, 20, 29, …
10

(2 + 9i )
i1
a1 
a10 
Questions 10 – 11. Find each sum if it exists. Leave answers as reduced fractions.

10.

 21.4
i 1
11.
i 1
a1 
 2.4
i 1
i 1
r=
_______
a1 
r=
_______